scholarly journals Multilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions

2018 ◽  
Vol 75 (3) ◽  
pp. 883-899 ◽  
Author(s):  
Yangzhang Zhao ◽  
Qi Zhang ◽  
Jeremy Levesley
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Tensor Product ◽  
Sparse Grids ◽  
Basis Functions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Smooth Basis

scholarly journals A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 964 ◽  
Author(s):  
Zhiyong Liu ◽  
Qiuyan Xu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Radial Basis Functions ◽  
Computational Cost ◽  
Basis Functions ◽  
Partial Differential ◽  
Compactly Supported ◽  
Linear Partial Differential Equations ◽  
Numerical Experimentation ◽  
Radial Basis

In this paper, we derive and discuss the hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations. Similar to multilevel splitting of finite element spaces, the hierarchical radial basis functions are constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the compactly supported radial basis functions approximation and stationary multilevel approximation, the new method can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system. These observations are obtained by taking the direct approach of numerical experimentation.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

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